On the Krull dimension of rings of integer-valued rational functions

TL;DR

This paper investigates the Krull dimension of rings of integer-valued rational functions over integral domains, especially focusing on Jaffard domains and PVDs, providing new insights and illustrative examples.

Contribution

It offers new results on the Krull dimension of integer-valued rational function rings over specific classes of domains, expanding understanding in this area.

Findings

Established bounds for Krull dimension in specific domains

Provided examples illustrating the theoretical results

Extended previous work on integer-valued polynomial rings

Abstract

Let $D$ be an integral domain with quotient field $K$ and $E$ a subset of $K$. The \textit{ring of integer-valued rational functions on} $E$ is defined as $$\mathrm{int}_R(E,D):=\lbrace \varphi \in K(X);\; \varphi(E)\subseteq D\rbrace.$$ The main goal of this paper is to investigate the Krull dimension of the ring $\mathrm{int}_R(E,D).$ Particularly, we are interested in domains that are either Jaffard or PVDs. Interesting results are established with some illustrating examples.